Relative symplectic cohomology and ideal-valued measures

Abstract: In a joint work in progress together with A. Dickstein, Y. Ganor, and L. Polterovich we prove new symplectic rigidity results. First, we categorify the notion of a heavy subset of a symplectic manifold (due to Entov-Polterovich), and in particular provide a simple algebraic criterion which guarantees that two heavy sets intersect. Next, we treat involutive maps defined on a symplectic manifold M; a smooth map M -> B is involutive if pullbacks of smooth functions on B Poisson commute. For such maps we prove a refinement of Entov-Polterovich's nondisplaceable fiber theorem, as well as a symplectic Tverberg-type theorem, which roughly says that each involutive map into a manifold of sufficiently low dimension has a fiber which intersects a wide family of subsets of M.

All of these results are proved using a generalized version of Gromov's notion of ideal-valued measures, which furnish an easily digestible way to package the relevant information. We construct such measures using relative symplectic cohomology, an invariant recently introduced by U. Varolgunes, who also proved the Mayer-Vietoris property for it, on which our work relies in a crucial manner. Our main technical innovation is the relative symplectic cohomology of a pair, whose construction is inspired by homotopy theory.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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